Orbital Exchange Calculations: An Alternative View of the Chemical Bond Paul B. Merrithewa PO Box 120, Amherst, New Hampshire 03031-0120 USA
Abstract The purpose of this paper is to explore a model of the chemical bond which does not assume that the electrons of the chemical bonding electron pair can be unambiguously identified with either the left hand or right hand of the bonding atoms when their orbitals overlap to bond. In order to provide maximum flexibility in the selection of the electron’s orbitals, the orbitals have been represented as spatial arrays and the calculations performed numerically. This model of the chemical bond assumes that the identifiability of the bonding electrons is a function of 1-(overlap/(1+overlap)) where the overlap of the two bonding electron’s orbitals is calculated in the usual manner. The kinetic energy of the bonding electron pair and the energy required to meet the orthogonality requirements, mandated by the Pauli principle, are a function of overlap/(1+overlap). The model assumes that the bonding orbitals are straight-forward atomic orbitals or hybrids of these atomic orbitals. The results obtained by applying this simple approach to eleven di-atomics and seven common poly-atomics are quite good. The calculated bond lengths are generally within 0.005Å of the measured values and bond energies to within a few percent. Bond lengths for bonds to H are about 0.02 Å high. Except for H2, bond lengths are determined, independent of bond energy, at that point where overlap/(1+overlap) equals 0.5.
I. INTRODUCTION The purpose of the work described here is to test a model for the chemical bond that assumes that the two electrons of the bonding pair are not completely identifiable, with respect to their source atom, when their orbitals overlap. Were the bonding electrons not identifiable in the overlap region it would be impossible to unambiguously identify every element of the orbital’s distribution with either the atom on the right hand or the atom on the left side of the bond. Since the identification is ambiguous, nature would allow one to make the most energy favorable identification, which is to picture the electrons spread out over the whole bond when calculating the kinetic energy. Distributions are representations of order (as differentiated from disorder or randomness). Narrow distributions represent high order, broader distributions less order. When two distributions overlap the order associated with the system decreases with their overlap because, in the overlap region, one can no longer identify each distribution element with a particular distribution. Overlap introduces randomness. Electron kinetic energy relates to order and electron distributions to representations of order. 1 In his early work applying LCAO-MO theory to H2, Coulson assumed that the two electrons of the bonding pair were identifiable. In subsequent applications of LCAO-MO theory to this molecule and more complex molecules, the same assumption has been made2. This assumption apparently has its origin in the view that a single electron can be uniquely assigned to one of two non-orthogonal overlapping spatial orbitals of different spin.
a Electronic mail: [email protected]
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That the electrons of two overlapping orthogonal spatial orbitals of the same spin are not distinguishable is manifest in the so-called “exchange integral”3. The exchange integral evaluates the electron-electron repulsion in the overlap region. The total electron-electron repulsion is reduced by this quantity (Since an electron cannot repel itself.). Likewise, it is reasonable to hypothesize that the electrons of the bonding pair, which have non-orthogonal spatial orbitals of different spin, are also not distinguishable in the overlap region.
A. Numerical Methods
The calculations described herein are done numerically with the orbitals represented in huge identical spatial arrays of the form phi[i][j][k] where i is the index for the radius from the molecular axis, j represents the distance along the molecular axis and k is an array identifier. (Only two spatial coordinates are needed since the electron structure is axially symmetric or can be treated as such.) In a numerical calculation, integrals become summations. For example, the electron-electron repulsion becomes: electron-electron repulsion = constant 퐼푀퐴푋 퐽푀퐴푋 퐼푀퐴푋 퐽푀퐴푋 ∑푖=0 ∑푗=0 ∑푙=0 ∑푚=0 phi[i][j][left array] phi[l][m][right array] ovr[ndist][i][l] , (1) where ovr[ndist][i][l] is the reciprocal of distance between phi[i][j][left array] and phi[l][m]{right array] and ndist = absolute value(-j+m). The value of the constant depends on the dimensions associated with the array elements. As a practical matter, for the repulsion calculation, adjacent phi[i][j] and phi[l][m] are consolidated to reduce the number of array elements. The electron-electron repulsion calculation requires less resolution than the kinetic energy calculation described below. Numerical methods are advantageous because they do not presume a particular functional form for the orbitals. These numerical methods are particularly advantageous in making bonding orbitals which are orthogonal to the opposite core 1s electrons. Numerical methods permit the use of iterative analytical methods to find the lowest energy orbitals which meet the orthogonalization requirements.
B. Kinetic Energy of Combined Orbital
Kinetic energy (KE) of an orbital is determined in the usual manner: KE = ∫∫2 2 r dr dl, (2) where represents a generic orbital and 2 is the Laplacian. The variable r represents the radial distance from the bond axis and l represents the distance along the bond axis. Using numerical methods this becomes: 퐼푀퐴푋 퐽푀퐴푋 KE =constant ∑푖=0 ∑푗=1 ((rrdn[i] (phi[i − 1][j] − phi[i][j]) − rrup[i](phi[i][j] − phi[i + 1][j]))) phi[i][j] + 퐼푀퐴푋 퐽푀퐴푋 constant ∑푖=0 ∑푗=1 ((phi[i][j + 1] − phi[i][j]) − (phi[i][j] − phi[i][j − 1])) (phi[i][j] rr[i]) , (3) where rr[i] is the axial radius at i
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rrdn[i] is one-half unit down the axial radius from rr[i] rrup[i] is one-half unit up the axial radius from rr[i]. (In the equation above i=0 leads to phi with a negative index (and therefore not determined). This is of no consequence because its factor, rrdn[i] = 0 when i=0.) The value of the constant depends on the dimensions associated with the array elements. As a practical matter, to increase accuracy while not sacrificing processing speed, finer arrays have been created for areas close to the nucleus (where phi changes faster) and coarser arrays at a distance from the nucleus. Another important quantity is the overlap of the bonding orbitals which is calculated in the usual manner: overlap = ∫∫l r 2 r dr dl, (4) where l represents the bonding atomic orbital on the left and r the bonding atomic orbital on the right. The orbitals l and r are atomic orbitals which have been made orthogonal to the core electrons on the opposite atom. Overlap has a range from 0.0 to 1.0. The atomic orbitals which have been made orthogonal to the core electrons on the opposite atom have no net overlap with those core-electron orbitals:
∫∫l 1sr 2 r dr dl =0.0 and (5)
∫∫r 1sl 2 r dr dl =0.0. (6) To calculate the kinetic energy when the identification of components of the orbitals as right versus left is ambiguous, one creates a hypothetical orbital, referred to as the combined orbital, l+r which distributes the electron equally on the atoms left and right of the bond while preserving the electron density map of the original left and right atomic orbitals combined and meeting the orthogonality requirements. For each spatial array element one computes: phi[i][j][combined] = square root (phi[i][j][right_ortho] phi[i][j][right_ortho]+phi[i][j][left_ortho] phi[i][j][left_ortho]) . (7) When phi[i][j][left_ortho] and phi[i][j][right_ortho] are negative, a negative sign is given to the combined orbital. The array elements phi[i][j][left_ortho] and phi[i][j][right_ortho] are the array elements of atomic orbitals which have been altered to create orbitals which are orthogonal to the core electrons of the opposite atom (l and r). The analytic procedure used to convert the set of array elements, phi[i][j][left] and phi[i][j][right], the original atomic orbitals elements, designated as l and r, to the set phi[i][j][left_ortho] and phi[i][j][right_ortho] is described in Section IIIA below. Note that the combined orbital is not the sum of the right and left orbitals as this would change the electron density map by adding electron density between the atoms. The author knows of no explicit functions which describe the set phi[i][j][left_ortho] and phi[i][j][right_ortho] or l+r. These calculations can only be performed numerically. The kinetic energy change associated with overlap of the atomic orbitals is designated KEbond. KEbond is calculated: KEbond = (overlap/(1+overlap)) KEnet (8)
where KEnet=(KEl+r - KEl - KEr) (9)
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The total kinetic energy change for the bond is 2.0 times KEbond. (KEbond is for one electron.) This approach to the kinetic energy assumes that, to the extent of overlap/(1+overlap), the electrons are spread over the bonding orbitals of both of the bonding atoms. To the extent of 1- (overlap/(1+overlap)), the kinetic energy is that of the atomic orbitals which have been made orthogonal to the opposite core electrons (the set phi[i][j][left_ortho] and phi[i][j][right_ortho]).
C. H2
Applying this approach to find the kinetic energy of the H2 bonding electrons, I find a bond energy, De, of 4.06 eV at a bond length of 0.753Å using an orbital scale factor (orbital reduction factor) of 1.15. (See column B of Table I for the results of this calculation at 0.754 Å.) The actual values are De equal to 4.75 at a bond length of 0.741 Å. Some improvement can be made by polarizing the H 1s orbitals by adding a small amount of 2pz (0.008) with a 2pz scale factor of 3.50 to the H 1s orbital while retaining an orbital scale factor of 1.15. This results in a bond energy, De, of 4.67 eV at a bond length of 0.754Å. (See column C of Table I for the results of this calculation at 0.754 Å.) These are remarkably good results for such a simple approach, but since the H2 results could be fortuitous, the investigation needs to be extended to other molecules. For purposes of comparison, Coulson’s early MO calculations have been reproduced here. These results are shown in Table II. Coulson’s MO approach gives poorer agreement with experiment, with De, of 3.49 eV. Coulson gives a breakdown of similar calculations for H2 at 1.41 Bohr (0.746Å) in his later book4. These are accurate to only about 0.1 eV. Coulson gives 17.8 eV, 19.3 eV, 27.2 eV and 16.2 eV for the electronic repulsion, nuclear repulsion, energy of two H atoms, and “energy of each separate molecular orbital”, respectively. My similar calculations for the same quantities at 1.41 Bohr give 17.824 eV, 19.299 eV, 27.220 eV and 16.095 eV.
Table I. Components of H2 calculations at 0.754 Å (Energy in electron volts)
A. Component B. Unpolarized 1s Orbital C. Polarized 1s Orbital orbital factor= 1.15 = fsb 1s+fpb 2pz fpb fpb=0.008 2pz orbital factor=3.50 1s orbital factor= 1.15 Kinetic energy gain associated with sigma orbital 4.1416 4.1365 overlap (2 KEbond) Nuclear-nuclear repulsion energy -19.0957 -19.0957
Nuclear to opposite bonding orbital energy 34.3914 36.8282
Electron-electron repulsion energy -14.7743 -16.0927
Energy for H atom compression/polarization -0.6056 -1.1023
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Total Energy 4.06 4.67
Table II. Reproduction of Coulson’s MO H2 calculation at 1.40 Bohr or 0.741 Å (Energy in electron volts)
Component Contribution to total energy Total kinetic energy of molecular orbitals -30.8357
Nuclear-nuclear repulsion energy -19.4367
Nuclear to molecular orbitals potential energy 98.8386
Electron-electron repulsion energy -17.8501
Energy of two H atoms -27.2259 orbital factor=1.197
Total 3.490
Note: Actual energy minimum is at 0.731 Å
III. ATOMIC ORBITALS AND ORBITAL SCALE FACTORS For the calculations described here, the atomic orbitals of Duncanson and Coulson5 have been used. The quantities Duncanson and Coulson call and c (These are equivalent to the effective nuclear charge [or orbital scale factor] for 2s and 2p electrons.), are increased by factors (called fact herein) ranging from 1.0 to 1.055. Contracting an atomic orbital raises an atom’s energy but enables a stronger bond. Table III gives the approximate energy impact of these small changes in the atomic parameters. The energy impact of these changes is usually less than 0.4 eV and is spread among several bonds in a polyatomic molecule. The factors are usually not difficult to establish since the energy increases exponentially as the factor is increased. (The orbital scale factor for C+ is taken as 1.10 and that for O- as 0.93.) Table II. Estimated Energy (eV) Associated with Orbital Scale Factor
Factors Atom 1.01 1.02 1.03 1.04 1.05 1.055 B — 0.01 0.05 0.09 0.15 C 0.01 0.04 0.11 0.20 0.33 0.44 N 0.03 0.11 0.22 0.41 O 0 0.10 0.27 0.54 F 0.04 0.20 0.50
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III. ORTHOGONALITY For calculations involving multi-electron atoms, orthogonality requirements must be considered. Consistent with the Pauli principle, the orbitals of electrons of the same spin must be orthogonal. When an atom presents a bonding orbital to the opposite atom two types of changes must be made to meet the orthogonality requirements: the bonding orbital must be made orthogonal to the “core” electrons on the opposite atom and the not-bonding electron orbitals must be changed/reconfigured to be orthogonal to the bonding orbital from the opposite atom.
A. Core Orthogonality
Herein, a bonding atomic orbital is made orthogonal to the opposite core by placing a node(s) in the orbital at the position of the nodes in the opposite bonding orbital. Since these calculations are performed analytically, using arrays to represent the orbitals, the orthogonalization is relatively easy. The goal is to find an orthogonal orbital with the same charge distribution (and potential energy) as the original atomic orbital. An orthogonal orbital with exactly the same charge density as the original atomic orbital is impossible because this would require a discontinuity in the function at the node. However, an iterative analytical procedure produces a satisfactory approximation of an orthogonal orbital, with approximately the same charge density as the original atomic orbital, but without a discontinuity. An approximate, and normalized orbital is constructed from the atomic orbital which has a relatively gentle transition at the node. The approximation procedure does not permit a displacement of charge density from one side of the nucleus to another or cause a net displacement of charge toward, or away from, the nucleus. The bond energy is calculated for successively sharper node transitions. [Alternatively, just the difference KEbond - (KE - KE) could be calculated]. Usually, as the transition becomes sharper, the bond energy will improve slowly, and the overlap change little. At some point the bond energy will decrease. The orbital that has the best energy is utilized. Sometimes, usually when one or both of the bonding atoms are “soft” (H, light elements), the bond energy does not improve as the node transition becomes sharper. In these cases the bond energy decreases slowly as the node transition sharpens. At some point the bond energy will begin to deteriorate more quickly for a given change in the node transition. The function just prior to the inflection point is chosen. Although the selection of the orthogonal bonding orbital is sometimes not precise, the process does not appear to introduce an error of more than a few percent even in the worst cases. There is a reason that the bond energy is relatively stable with changes in the bonding orbital. As the node transition sharpens, KE increases. The increase in KE is accompanied by a corresponding increase in KEl+r. The difference between these two quantities remains relatively constant. Making the bonding atomic orbital orthogonal to the opposite core has the salubrious effect of making the two overlapping bonding orbitals positive where the other is positive, and negative where the other is negative. The combined orbital, l+r has no discontinuities.
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B. Valence Orthogonality
When an atom presents a bonding orbital to the opposite atom, the opposite atom must change/reconfigure so that its orbitals, with the exception of its bonding orbital, are orthogonal to the bonding orbital. This is referred to here as valence electron orthogonalization. Consider atoms in di-atomic molecules. The atoms in di-atomic molecules typically have an s2pn n-1 configuration. To orthogonalize, the atoms reconfigure to spzp spo where spo is an opposing hybrid orbital (taking z as the bond axis). The 2s and 2pz orbitals of the atom hybridize to form an opposing orbital of the form fso 2s - fpo 2pz, (fso stands for fraction s opposing) and a bonding orbital of the form fsb 2s + fpb 2pz (fsb stands for fraction s bonding),where fsofso + fpofpo=1. The coefficients fso and fpo are chosen to make the opposing orbital orthogonal to the opposite bonding orbital. The opposing orbitals determine the bonding orbitals (fsb = fpo and fpb = fso). In the case of a di-atomic, fsor (the subscripts r and l indicate right and left) and fpor and fsol and fpol are chosen to satisfy the simultaneous equations:
fsbl fsor overlap s_s_n - fpbl fpor overlap pz_pz_n + fsor fpbl overlap pz_s_n - fpor fsbl overlap s_pz_n = 0.0 and (10a)
fsbr fsol overlap s_s_n - fpbr fpol overlap pz_pz_n + fsal fpbr overlap pz_s_n - fpal fsbr overlap s_pz_n = 0.0 . (10b)
The overlap s_s_n is the overlap of the 2s atomic orbitals (not core orthogonalized) and etc.
C. Orthogonality Energy
The orthogonalizations described above in Section IIIA need only be taken to the extent that the bonding atomic orbitals are identifiable (when the orbitals are identifiable the bond is said to be “not bonding”). The measure of identifiability (bond is “bonding”) is termed fraction_bonding and is calculated: fraction_bonding = overlap/(1.0+overlap). (11)
The energy required to core orthogonalize a bonding orbital, KEcore_ortho, is:
KEcore_ortho = (1.0-fraction_bonding) (KE - KE) (12) where KE is the kinetic energy of the core orthogonalized atomic orbital and KE is the kinetic energy of the original atomic orbital. For two hybrid sigma bonding orbitals (comprised of the 2s and 2pz atomic orbitals) of the form r= fsbr 2s+fpbr 2pz and l= fsbl 2s+fpbl 2pz, the energy to core orthogonalize the hybrid orbitals, KEcore_ortho, is:
KEcore_ortho = (1-fraction_bonding) (( fsbl fsbr fsbl fsbr (KEs_sal - KEsl) + fsbl fsbr fsbl fsbr (KEs_sar - KEsr) + fsbl fpbr fsbl fpbr (KEs_pzal - KEsl) + fsbl fpbr fsbl fpbr (KEpz_sar - KEpzr) + fpbl fsbr fpbl fsbr (KEpz_sal - KEpzl) + fpbl fsbr fpbl fsbr (KEs_pzar - KEsr) +
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fpbl fpbr fpbl fpbr (KEpz_pzal - KEpzl) + fpbl fpbr fpbl fpbr (KEpz_pzar - KEpzr)) (13) where KEsl and KEpzl are the kinetic energies of the left 2s and 2pz initial atomic orbitals and, KEsr and KEpzr are those on the right. KEs_sal is the kinetic energy of the s orbital on the left which has been orthogonalized with the right hand s (the subscript “a” indicates the orthogonalized orbital). KEs_pzal is the kinetic energy of the s orbital on the left which has been orthogonalized with the right hand 2pz. KEpz_sal is kinetic energy of the 2pz orbital on the left which has been orthogonalized with the right hand s. KEpz_pzal is kinetic energy of the 2pz orbital on the left which has been orthogonalized with the right hand pz. The suffix r indicates the corresponding orbitals on the right. In the potential energy calculations, the reconfiguration required to meet the valence orthogonalization requirement need only to be taken to the extent of (1.0-fraction_bonding). If a 2s orbital is reconfigured to a fso 2s-fpo 2pz hybrid to become orthogonal, then, in the potential energy calculations, the electron is considered to be in an 2s orbital for fraction_bonding and in a fso 2s-fpo 2pz hybrid orbital for (1.0-fraction_bonding). In this latter case, the bond energy is also adjusted for the energy required to promote the s to p. If the energy to promote an s to p is given by stop, then the bond energy is decreased by stop fpo fpo (1.0-fraction_bonding). A table of the energies required for the 2s to 2p promotion for the first row atoms has been developed for this work. The stop values are in the order BC Table IV. Estimated Energy to Promote 2s to 2p (stop) Values Species 2p Orbital Type stop Energy/eV B spo (unpaired) 4.35 B spo (paired) 6.55 C+ spo (unpaired) 4.35 C+ p 5.05 C Td 6.55 C spo (unpaired) 6.55 C p (unbonded) 6.95 C p 7.50 C spo (paired,bonding) 7.90 N spo (paired) 12.40 O spo (paired) 16.60 O- spo (paired) 16.60 8 D. Core Orthogonality of a Second Sigma Pair Most atoms in di-atomic molecules and some atoms in other molecules have more than one bond pair of electrons with sigma symmetry. For example consider N2. N has the 2 2 3 2 2 2 configuration 1s 2s 2p or 1s 2s 2p 2pz. There are two possibilities for sigma bonds: an sp hybrid to sp hybrid bond and an s to s bond. To the extent that one pair (or both pairs) of the bonding electrons are not identifiable, (fraction_bonding), the second 2s electron remains in place. To the extent that this s to s bond pair is not bonding (1-fraction_bondingsecond_set), the second 2s must also be made orthogonal to the opposite core electrons. The energy to core orthogonalize a second sigma orbital KEcore_ortho_x, is, in general, for the left: KEcore_ortho_xl = fraction_bonding (1.0-fraction_bondingsecond_set) (fsbl fsbr fsbl fsbr (KEs_sal - KEsl) + fsbl fpbr fsbl fpbr (KEs_pzal - KEsl) + fpbl fsbr fpbl fsbr (KEpz_sal - KEpzl) + fpbl fpbr fpbl fpbr (KEpz_pzal - KEpzl)). (14) The calculations for the right side are analogous to those on the left. As (12) indicates the KEcore_ortho_x penalty is taken only to the extent of overall fraction_bonding. If there are second sigma bonding orbitals on both sides of the bond and these bond, then the KEcore_ortho_x penalty is further reduced to the extent that the second set of sigma orbitals themselves bond, fraction_bondingsecond_set. IV. SIGMA HYBRID ORBITAL OVERLAP AND KINETIC ENERGY For a single set of hybrid sigma bonding orbitals of the form r= fsbr 2s + fpbr 2pz and l= fsbrl 2s + fpbl 2pz ,the quantities overlap and KEbond are: overlap = fsbl fsbr overlap s-s + fpbl fpbr overlap pz-pz + fpbl fsbr overlap pz-s + fsbl fpbr overlap s-pz and (15) KEbond = (1.0/(1.0+overlap)) (fsbl fsbr overlap s-s KE net s-s+ fpbl fpbr overlap pz-pz KE net pz-pz + fpbl fsbr overlap pz-s KE net pz-s+ fsbl fpbr overlap s-pz KE net s-pz). (16) Obviously, overlap cannot exceed 1.0, nor can fraction_bonding exceed 0.5. Except for the bonds of some “soft” atoms (H typically) fraction_bonding generally reaches the limit of 0.5. The limitation on overlap makes it possible, in most cases, to determine bond lengths without a complete treatment of bond energy. With a few exceptions, bond energy increases as the bond length decreases until fraction_bonding = 0.5 is reached. V. DUAL AND PARALLEL BONDS As mentioned above, most atoms in di-atomic molecules and some atoms in other molecules have more than single pair of electrons that have the appropriate symmetry for sigma 9 bonding. Both of these pairs bond. This results in what is described herein as dual bonding or parallel bonding. Dual/parallel bonds are not double bonds in the traditional sense, i.e a sigma bond and a pi bond. Dual/parallel bonds entail two sigma bonds. Parallel bonding differs from dual bonding in that the second set of bonding orbitals does not reconfigure when the bond is not bonding. Dual bonding is far more common than parallel bonding. Parallel bonding is exhibited + in F2, He2 and partially in other compounds. A. Dual Bonding The calculation of dual overlap, fraction_bonding and KEbond differs somewhat from that of the single sigma bond because simultaneous overlaps must be taken in account. 2 Consider two atoms each with a sigma bonding configuration of 2s 2pz (e.g. N2). The dual bond between these atoms could be considered as an sp-sp hybrid bond and an s-s bond with each weighted by 0.5. Alternatively, the dual bond between these atoms could be considered as an sp-sp hybrid bond, an s-s bond, an sp-s bond and an s-sp bond with each 2 weighted by 0.25. In the latter case (s pz on both sides), the relevant quantities are: fraction_bonding s-s = overlap s_s/(1.0+overlap s_s) (17a) fraction_bonding s-sp = overlap s_sp/(1.0+overlap s_sp) (17b) fraction_bonding sp-s = overlap sp_s/(1.0+overlap sp_s) (17c) fraction_bonding sp-sp = overlap sp_sp/(1.0+overlap sp_sp) (17d) fraction_bonding ave = 0.25 (fraction_bonding s-s + fraction_bonding s-sp + fraction_bonding sp-s + fraction_bonding sp-sp) (18) overlap ave = fraction_bonding ave/(1.0-fraction_bonding ave) (19) simultaneous_bond s-s/sp-sp = fraction_bonding s-s fraction_bonding sp-sp (20a) simultaneous_bond s-sp/sp-s = fraction_bonding s-sp fraction_bonding sp-s (20b) overlap s-s/sp-sp = simultaneous_bond s-s/sp-sp /(1.0- simultaneous_bond s-s/sp-sp) (20a) overlap s-sp/sp-s = simultaneous_bond s-sp/sp-s/(1.0- simultaneous_bond s-sp/sp-s) (20b) overlap = 2 overlap ave - overlap sp-s/s-sp - overlap s-sp/sp-s (21) fraction_bonding = overlap/ (1.0+overlap) factor = fraction_bonding/fraction_bonding ave (22) KEbond = factor 0.25 ( KEbond s-s + KEbond s-sp + KEbond sp-s + KEbond sp-sp) (23) 10 2 With the N2 bonding configurations, both sides 2s 2pz, the dual bond overlap is actually somewhat less than the overlap of the sp-sp overlap alone. This is because the s-s overlap is much less than the sp-sp overlap. 2 2 Consider two atoms each with a sigma bonding configuration of 2s 2pz (e.g. O2). In this case the relevant quantities are: fraction_bonding sp-sp = overlap sp_sp/(1.0+overlap sp_sp) (24) fraction_bonding ave = fraction_bonding sp-sp (25) overlap ave = fraction_bonding ave/(1.0-fraction_bonding ave) (26) simultaneous_bond sp-sp/sp-sp = fraction_bonding sp-sp fraction_bonding sp-sp (27) overlap sp-sp/sp-sp = simultaneous_bond sp-sp/sp-sp/(1.0- simultaneous_bond sp-sp/sp-sp) (28) overlap = 2 overlap ave – 2 overlapsp-sp/sp-sp (29) fraction_bonding = overlap/ (1.0+overlap) factor = fraction_bonding/fraction_bonding ave (30) KEbond = factor KEbond sp-sp (31) 2 2 With these bonding configurations, both atoms 2s 2pz , the dual bond overlap is somewhat more than the overlap of the sp-sp overlap alone. 2 Consider two atoms, one with a sigma bonding configuration of 2s 2pz and one with a 2 2 sigma bonding configuration of 2s 2pz (e.g. NO). In this case the relevant quantities are: fraction_bonding s-sp = overlap s_sp/(1.0+overlap s_sp) (32) fraction_bonding sp-sp = overlap sp_sp/(1.0+overlap sp_sp) (33) fraction_bonding ave = 0.5 (fraction_bonding s-sp + fraction_bonding sp-sp) (34) overlap ave = fraction_bonding ave/(1.0-fraction_bonding ave) (35) simultaneous_bond s-sp/sp-sp = fraction_bonding s-sp fraction_bonding sp-sp (36) overlap s-sp/sp-sp = simultaneous_bond s-sp/sp-sp/(1.0- simultaneous_bond s-sp/sp-sp) (37) overlap = 2 overlap ave – 2 overlap s-sp/sp-sp (38) fraction_bonding = overlap/ (1.0+overlap) factor = fraction_bonding/fraction_bonding ave (39) KEbond = factor 0.5 (KEbond s-sp + KEbond sp-sp) (40) B. Parallel Bonding 11 Parallel bonding is a type of dual bonding that occurs when, instead of reconfiguring to orthogonalize, a second sigma orbital remains in place and forms a node to make it orthogonal to the opposite side bonding orbital. The best example of parallel bonding is F2. F has a bonding 2 2 2 3 configuration of 1s 2s 2pz 2pxy . For parallel bonding, fraction_bonding is simply the sum of fraction_bonding for each of the two sigma bonds. So for the F2 parallel bond: fraction_bonding = fraction_bonding sp-sp + fraction_bonding sp-sp and (41) KEbond = KEbond sp-sp + KEbond sp-sp . (42) Since fraction_bonding calculated in this manner is subject to the usual constraint (fraction_bonding 0.5) parallel bonding results in relatively long bond lengths and is usually associated with molecules which require p orthogonalization (e.g. F2)(See Section VI.C below). VI. PI BONDING, PI RESONANCE AND PI ORTHOGONALIZATION A. Pi Bonding The kinetic energy reduction associated with pi bonding, analogous to that of sigma bonding, is: KEbond_ = fraction_bonding (KEcombined_ – KE_l – KE_r) (43) where fraction_bonding = overlap/(1.0+overlap). (44) Pi overlaps are much smaller than sigma overlaps, usually in the range of 0.1 to 0.3. Pi bonding only occurs to the extent that both pi bonding orbitals have the appropriate symmetry. B. Pi Resonance When the bonding electron is free to move from the right bonding orbital to the left, or from the left to the right, fraction_bonding in (39) above is 0.5. The kinetic energy reduction associated with full pi resonance, KEbond__res, is: KEbond__res = 0.5 (KEcombined_ – KE_l – KE_r) (45) For these calculations, the energy associated with the formation of a positively charge species in a resonance is obtained from tables of ionization potentials6. The energy associated with the formation of negatively charged species is obtained from tables of electron affinities7. C. Pi Orthogonalization When the porbitals on the atom on one side of the bond are more than half full (three or four electrons) and the porbitals on the other side are at least half full, then, one of each of the spin paired pi orbitals must be made orthogonal to the pi bonding orbitals from the opposite side. This orthogonalization is analogous to that described in Section IIIA above with respect to sigma bonding, with a node placed in the porbital to make it orthogonal to the opposite pi bonding orbital. An analytical procedure, similar to the one used for core orthogonalization described 12 above, is used to find the optimal node position and node transition. As in core orthogonalization, this procedure attempts to maintain the atomic orbital density distribution as closely as possible. Analogous to sigma bonding, the pi orthogonalization penalty is taken only to the extent of (1.0-fraction_bonding). VII. SECONDARY BONDING In poly-atomic molecules, bonding occurs not only to the closest atom, but also to all atoms in the molecule with which it significantly overlaps. Secondary/tertiary bonding differs from primary bonding in three ways. First, the bond axis of the primary is retained in the secondary, tertiary, etc.. Secondly, secondary bonding is reduced by the extent of primary bonding (and the tertiary by the extent of primary and secondary and etc.). The reduction in secondary bonding, by primary bonding, is determined by the least primary bonding of the two secondary bonding orbitals. Thirdly, total overlap, including contributions from secondary and subsequent bonds, is calculated along the principle axis which is usually the primary bond axis. The secondary (tertiary, etc.) overlap between the primary bonding orbital on the left and the secondary bonding orbital on the right is as follows: overlapsec_l = fsbl fsbr overlaps-s+cosθl cosθr fpbl fpbr overlappz-pz+ cosθl fpbl fsbr overlappz-s+cosθr fsbl fpbr overlaps-pz , (46) where θl is the angle between the primary axis of the left atom and the secondary (tertiary, etc.) bond axis and where θr is the angle between the primary axis of the right atom and the secondary (tertiary, etc.) bond axis. Notice that fsbl and fpbl refer to the hybridization of the primary orbital on the left and fsbr and fpbr refer to the hybridization of the secondary orbital on the right. Note also that, in general, the secondary for the left is different from the secondary on the right. The calculation of overlapsec_r is analogous to that on the left. Sometimes there is a secondary on one side but none on the other (e.g. CH4). The overlap component along the primary bond axis, overlapsec_l_z, is: 2 overlapsec_l_z = cos θ overlapsec_l , (47) where θ is the angle between the primary bond axis and the secondary (tertiary, etc.) bond axis. The secondary fraction_bonding increment, fraction_bondingsec_l_inc, is: fraction_bondingsec_l_inc = 0.5 (1–fraction_bondinglessor_primary_l) (overlapsec_l_z/ (1+ overlapsec_l_z) (48) where fraction_bondinglessor_primary_l is the lessor of the two fraction_bondingprimary associated with the secondary. The factor of 0.5 arises because the secondary bond spans two primary bonds. The contribution of the secondary overlap to the total overlap, the secondary overlap increment, overlapsec_l_inc, is: overlapsec_l_inc = fraction_bondingsec_l_inc/ (1-fraction_bondingsec_l_inc). (49) 13 The calculations for the right side are analogous to those on the left. The overlap subtotal, overlapsubtotal, is: overlapsubtotal = overlapprimary+overlapsec_l_inc+overlapsec_r_inc, and (50) fraction_bondingsubtotal = overlapsubtotal/ (1+ overlapsubtotal). (51) Subsequent, secondary (or tertiary and etc.) are treated similarly. Summing overlap contributions in this manner is very important because they make up a very significant portion of the total overlap (typically around 10-25%) and therefore have a very large impact on the bond length and, indirectly, via the overlaptotal=1.0 constraint, on the bond energy. The direct secondary contributions to the bond energy (described below) can sometimes be ignored for an approximate result but the secondary contributions to overlap cannot be ignored. The core orthogonalization penalty associated with the overlap of the primary bonding orbital on the left and the core electrons of the secondary atom on the right is as follows: KEcore_ortho_sec_l = (fsbl fsbr fsbl fsbr (KEs_sal - KEsl) + fsbl fpbr fsbl fpbr (KEs_pzal - KEsl) + cos(θ) fpbl fsbr fpbl fsbr (KEpz_sal - KEpzl) + cos(θ) fpbl fpbr fpbl fpbr (KEpz_pzal - KEpzl)). (52) The calculations for the right side are analogous to those on the left. The core_orth contributions are taken only to the extent of (1-fraction_bondingtotal). Fraction_bondingtotal generally reaches 0.5. The KEcore_orth_sec contributions to the bond energy are generally quite small. Secondary potential energy terms are calculated in the same manner as those of the primary terms. The secondary and subsequent contributions are much smaller than the primary due to the longer atomic distances, but they are nonetheless usually significant. Table V provides a complete breakdown of the many components that contribute to the total overlap in diamond. Table V. Components of the Calculation of Overlap in Diamond at 1.541 Å 2 2 The configuration of C in diamond is 1s 2s2pz2pxy . Component Value Overlap Increment Sigma overlap of primary orbitals (overlap 1st) 0.7960 0.7960 overlap 1st = 0.25 overlap s-s + 0.25 overlap s-p+0.25 overlap p-s+ 0.25 overlap p-p fsbr = fsbl = fpbr = fpbl = 0.5 Fraction_bonding of primary (fraction_bonding 1st) 0.4432 fraction_bonding 1st = overlap 1st /(1+ overlap 1st) 14 Component Value Overlap Increment Sigma overlap to secondary along primary axis (overlap 2nd) 0.0949 3 secondary atoms are at (0,1/2,1/2) at 2.517 Å The angle from bond axis is 35.265 degrees. cos(35.265)=0.8165 overlap 2nd = 0.8165 0.8165 (0.25 overlap s-s+0.8165 0.25 overlap s-p+ 0.8165 0.25 overlap p-s+0.8165 0.8165·0.25 overlap p-p) Secondary fraction_bonding (fraction_bonding2nd) 0.0867 fraction_bonding2nd = overlap 2nd /(1+overlap 2nd) First secondary fraction_bonding increment (fraction_bonding2nd_inc) 0.0483 fraction_bonding2nd_inc = 2 0.5 (1- fraction_bonding 1st) fraction_bonding2nd Secondary fraction_bonding is reduced by previous, coincident, 1st order bonding. 0.5 because secondary spans 2 primary 2 because primary C is also a secondary First secondary overlap 0.0507 first secondary overlap = fraction_bonding2nd_inc/(1- fraction_bonding 2nd_inc) Total secondary overlap for 3 atoms 0.1479 0.1479 Note that coincident bonding increases as previous fraction_bonding increases with successive bonds. 0.1479<3 0.0507 Sigma overlap to tertiary #1 (overlap 3rd #1) 0.0108 6 tertiary atoms are at (1/4,3/4,-1/4) and (3/4,1/4,-1/4 ) at 2.9512 Å cos(angle)=0.5222 overlap 3rd #1 = 0.5222 0.5222 (0.25 overlap s-s+0.5222 0.25 overlap s-p+ 0.5222 0.25 overlap p-s+0.5222 0.5222 0.25 overlap p-p) Total tertiary overlap #1 for 6 atoms 0.0329 0.0329 Sigma overlap to tertiary #2 (overlap 3rd #2) 0.0089 3 tertiary atoms are at (1/4,3/4,3/4 ) at 3.8781 Å cos(angle)=0.9270 Total tertiary overlap #2 for 3 atoms 0.0134 0.0134 Sigma overlap to tertiary #3 (overlap 3rd #3) 0.0004 7 tertiary atoms are at opposite faces at 4.3592 Å (4 above unit cell) cos(angle)=0.4717 Total tertiary overlap #3 for 7 atoms 0.0013 0.0013 Sigma overlap to tertiary #4 (overlap 3rd #4) 0.0041 3 tertiary atoms are at adjacent corner at 3.5593 Å cos(angle)=0.5774 Total tertiary overlap #4 for 3 atoms 0.0061 0.0061 Sigma overlap to tertiary #5 (overlap 3rd #5) 0.0009 3 tertiary atoms are at one off from adjacent corner at 5.5372 Å cos(angle)=0.7360 Total tertiary overlap #5 for 3 atoms 0.0014 0.0014 15 Component Value Overlap Increment Sigma overlap to tertiary #6 (overlap 3rd #6) 0.0004 3 tertiary atoms are at corners at (1,1,0),(1,0,1)and (0,1,1) at 5.0336 Å cos(angle)=0.8167 Total tertiary overlap #6 for 3 atoms 0.0006 0.0006 Sigma overlap to tertiary #7 (overlap 3rd #7) 0.0000 3 tertiary atoms are at opposite corners at 6.165 Å cos(angle)=0.8167 Total tertiary overlap #7 for 3 atoms 0.0001 Total Overlap 0.9997 VII. RESULTS Some results from the application of the methods described herein are presented in Tables VI, VII and VIII. Table IX provides a description of the components of the C2 calculation. Table X is a detailed description of the components of the CH4 calculation. Bond lengths are generally accurate to 0.005Å and bond energies to a few percent. Except for a few bonds to hydrogen, the bond length results are independent of the bond energy results. Of these results, only for H2 is the bond length determined by a maximum in the bond length versus bond energy curve. Otherwise the bond length is found at the point where overlap becomes 1.0 (and fraction_bonding becomes 0.5) as the bond length is decreased. Aside from the character of the original atomic orbital, bond lengths are impacted only by core orthogonalization and the selected orbital scale factor. Finding the optimum parameters for the core orthogonalization, using the analytical procedures described above, particularly for C through F, is not difficult. Bond lengths are not sensitive to small changes from the optimal orthogonalization parameters although care must be taken in the core orthogonalization to assure accurate bond energies. Orbital scale factors are generally easy to establish as the energy required to compress an atomic orbital rises rapidly with the factor. The factors used are shown in the results tables for comparison purposes. Consistent with the data in Table I, the factors are larger for lighter elements. Orbital scale factors for poly-coordinate atoms are larger than those of di-atomics because the energy to compress the orbital is spread among several bonds. Bond energies are sensitive to the magnitude of the stop value. Care has been taken to use the same stop value in different compounds in which the atom shows a similar electronic structure. The stop values used for each bond are shown in the results tables for comparison purposes. The results tables provide the bonding and not bonding configurations of the bonding atoms. In these tables 2sp0 indicates an opposing orbital and 2sp is a hybrid orbital made by adding a small amount of 2p to a 2s orbital. The z axis is taken as the bond axis. Notice that in - O2 and O2 , the bonding configurations include two 2pz. One of these is reconfigured as 2p (or more precisely, 2p since there is pi bonding here) in the not bonding configuration. In CH3 and graphite, one of the C atom’s 2s orbitals is promoted to 2p when any of the central C atom’s 3 bonds is not bonding [(1.0-0.5 0.5 0.5)=0.875] to meet the orthogonality requirement. In graphite, this promotion allows pi bonding when the C atoms on both sides of the bond are not 16 bonding (0.875 0.875). In graphite pi bonding is reduced by the extent of simultaneous pi bonding to other Cs. Hybridization of the bonding orbitals on central atoms in poly-atomics is different from that in di-atomic molecules or terminal atoms. The hybridization of bonding orbitals on atoms that are not in the terminal position is determined via the availability of s character. In three- coordinate CH3 (D3h symmetry) and graphite, the square of the central C bonding orbital s coefficient, fsb fsb, is 0.667. The s character in the bonding orbital cannot be oversubscribed, so 3 fraction_bonding fsb fsb must be less than or equal to 1.0 and, since fraction_bonding=0.5, fsb fsb=0.667. The s character in a bonding orbital is maximized because this leads to a lessor orbital overlap and a shorter bond. Following the same logic, four-coordinate atoms or pseudo four-coordinate central atoms (those which have a combination of four lone pairs and bonds) have fsb fsb=0.5. Diamond and methane have four-coordinate central atoms and have Td symmetry. Water and ammonia are pseudo four-coordinate. Two-coordinate atoms such as CO2 have fsb fsb= 0.75. In the two-coordinate case, the s character of the central atom’s orbital is limited because, when both sides are not bonding, 0.25 of the time, fsb fsb must equal 0.5. 3 Consistent with the traditional view, NH3 and H2O have one and two traditional sp orbitals containing lone (non-bonding) pairs of electrons respectively. The bond axes in these molecules are bent inward relative to the electronic axes to make the lone pair(s) orthogonal to the H 1s bonding orbital. The calculated bond angles reflect the requisite bending. Dipole moments, , have been calculated for NO, NH3 and H2O. Dipole moments are very sensitive to the electronic structure of the molecule. The calculated values are in good agreement with the observed values. Table VI. Results for Some Di-Atomics Which Exhibit No Resonance a b Molecule De or D0 /eV Bond Configurationsc,d,e,f Bonds stop/ Length/Å (C to O 1s2 omitted) eV H2 observed observed H 1s(polarized w/pz) 4.75 0.741 H fsb·fsb=0.992 σ n/a calculated calculated fact1s=1.15 4.67 0.754 fact2pz=3.5 2 2 C2 observed observed bonding C 2sp 2p 2 σ (dual) 6.3 1.2425 2 not bonding 2spo2sp2p fsbfsb=0.957 6.55 calculated calculated fact2s,2p=1.035 2 (unpaired) 6.51 1.243 2 2 N2 9.91 1.0977 bonding 2s 2pz2p 2 σ (dual) 2 calculated calculated not bonding 2spo2s2pz2p 2 12.40 9.80 1.100 fact2s,2p=1.015 2 BN bonding B 2s 2pz observed observed not bonding B 2spo2s2pz 2 σ (dual) B 6.55 3.92 1.281 2 2 bonding N 2s 2pz2p⊥ no N 12.40 calculated 0.08 not bonding N calculated 1.281 2 2spo2s2pz2p⊥ 4.00 fact2s,2p(B,N)=1.04,1.015 17 a b Molecule De or D0 /eV Bond Configurationsc,d,e,f Bonds stop/ Length/Å (C to O 1s2 omitted) eV 2 2 CN bonding C 2sp 2p observed observed 2 2 bonding N 2s 2pz2p 2 σ (dual) C 6.55 7.78 1.172 2 not bonding C 2spo2sp2p C fsbfsb=0.953 (unpaired) 0.03 calculated not bonding N 2 N 12.40 calculated 1.169 2 7.67 2spo2s2pz2p fact2s,2p(C,N)=1.035,1.01 2 2 NO observed bonding N 2s 2pz2p 6.56 observed 2 2 3 bonding O 2s 2pz 2p 2 σ (dual) N 12.40 1.1506 calculated not bonding N 2 O 16.60 calculated 6.42 2 2spo2s2pz2p 1.154 not bonding O (calc=0.133 D 2 2spo2s2pz2p 2pa expt=0.159 D) fact2s,2p(N,O)=1.02,1.02 a Homonuclear di-atomics. De is the equilibrium dissociation energy which is slightly larger than the molecular ground-state-dissociation energy D0. Observed data are from Ira N. Levine, Quantum Chemistry 7th edition, pp 373, Pearson (2014) and/or John P. Lowe, Quantum Chemistry 2nd edition, pp 224, Academic Press (1993). b Heteronuclear di-atomics. Observed D0 data are from W.M. Hayes, Editor in Chief, CRC Handbook of Chemistry and Physics, Section 9, CRC Press (2014). c fact is the orbital scale factor. d 2sp is a hybrid orbital made by adding a small amount of 2p to the existing 2s orbital. e 2spo is a hybrid opposing orbital formed to meet the valence orthogonalization requirement. f 2pa is 2p made orthogonal to opposite 2p. Table VII. Results for Some Di-Atomics Which Exhibit a Full Pi Resonance a Molecule De /eV Bond Configurationsb,c,d,e Bonds stop/ Length/Å (1s2 omitted) eV 2 B2 observed observed bonding B 2sp 2p 2 3.1 1.590 bonding B+ 2sp 2 σ (dual) calculated calculated 2 2 bonding B- 2sp 2p fsbfsb=0.953 4.35 3.21 1.582 not bonding B+,B,B- resonance (unpaired) 0,1,2 2spo2sp2p fact2s,2p=1.035 + 2 2 C2 observed observed bonding C 2sp 2p 2 σ (dual) C 6.55 2 5.3 1.301 bonding C+ 2sp 2p fsbfsb=0.915 (unpaired) calculated calculated 1.2 not bonding 2spo2sp2p resonance C+ 4.35 5.22 1.301 fact2s,2p(C,C+)=1.08 (unpaired) O2 observed observed bonding O+,O,O- 2 σ (dual) 2 2 1,2,3 5.21 1.2074 2s 2pz 2p fsbfsb=0.40 16.60 calculated calculated not bonding O+,O,O- 2 resonance plus 5.25 1.208 2,2,2 0,1,2 2spo2s2pz2p 2pa plus when both fact2s,2p=1.0225 sides s⇒p 18 a Molecule De /eV Bond Configurationsb,c,d,e Bonds stop/ Length/Å (1s2 omitted) eV - O2 observed observed bonding O,O- 2 σ (dual) 2 2 2,2 0,1 4.14 1.32 2s 2pz 2p 2pa fsbfsb=0.46 O 16.60 calculated calculated not bonding O,O- resonance plus O- 16.60 4.11 1.28 2,2 1,2 2spo2s2pz2p 2pa 2 fact2s,2p(O,O-)=0.965 F2 observed observed bonding F 2 σ (parallel) 2 2 2 1.66 1.412 2s 2pz 2p 2pa fraction_bond=0.25 n/a calculated calculated not bonding F for each of two σ 1.74 1.418 2 2sp02s2pz2spa2p 2pa 2 resonance plus 2 fact2s,2p=1.01 a th De is the equilibrium dissociation energy. Observed data are from Ira N. Levine, Quantum Chemistry 7 edition, pp 373, Pearson (2014) and/or John P. Lowe, Quantum Chemistry 2nd edition, pp 224, Academic Press (1993). b fact is the orbital scale factor. c 2sp is a hybrid orbital made by adding a small amount of 2p to the existing 2s orbital. d 2spo is a hybrid opposing orbital formed to meet the valence orthogonalization requirement. e 2pa is 2p made orthogonal to opposite 2p. Table VIII. Some Poly-atomics a,b d Molecule Do /eV Bond Configurations Bonds stop/eV c Å /D Length / (1s2 omitted) c /° CH/CH3 observed observed all bonding C (0.125) σ 2 (methyl) 4.22 1.076 2s 2pz2p⊥ C fsbfsb=0.6667 6.95 calculated calculated 2 otherwise 2s2pz2p⊥ 4.25 1.080 fact2s,2p(C)=1.05 H polarized fact1s(H)=1.08 H fsb·fsb=0.977 CO/CO2 both bonding C,C+ σ (dual when opposite observed observed 2 1,0 2s 2pz2p bonding i.e. s⇏p) C 7.50 8.33 1.160 any not bonding C,C+ C fsbfsb=0.75 C+ 5.05 calculated calculated 2,1 2s2pz2p resonance 8.33 1.160 bonding O,O- [O-C+O,OCO,OC+O-] O 16.60 2 2 1,2 1,1 2s 2pz 2p 2pa bond (reduced for O- 16.60 not bonding O,O- coincident ) 2,3 1,1 2s2pz2spo2p 2pa when s⇒p (i.e. fact2s,2p(C,C+)=1.03,1.11 one/both not sigma fact2s,2p(O,O-)=1.01,0.93 bonding) CH/CH4 observed observed C σ 2 (methane) 4.30 1.087 2s2pz2p⊥ C fsbfsb=0.5 6.55 calculated calculated fact2s,2p(C)=1.05 H polarized 4.32 1.104 fact1s(H)=1.13 H fsbfsb=0.984 19 a,b d Molecule Do /eV Bond Configurations Bonds stop/eV c Å /D Length / (1s2 omitted) c /° NH/NH3 observed observed N σ (ammonia) 4.05 1.012 2s2pz N fsbfsb=0.565 12.40 expt HNH= 2+0.5 0.5 2p⊥ 2spo (fsbfsbis nominally 0.5 =1.471 106.7 axes bent to meet calculated orthogonality) 1.023 fact2s,2p(N)=1.01 calculated H polarized fact1s(H)=1.17 4.16 HNH= H fsbfsb=0.958 calc =1.47 106.8 OH/OH2 observed observed O σ 4.79 0.9575 1+0.25 (water) 2s 2pz O fsbfsb=0.66 16.60 expt=1.854 HOH= 3+0.25 0.5 2p⊥ 2spo (fsbfsbis nominally 0.5 104.5 axes bent to meet calculated calculated orthogonality) 0.923 fact2s,2p(O)=1.01 4.79 H polarized fact1s(H)=1.16 calc=1.83 HOH= H fsbfsb=0.92 104.5 2 CC/ 2s2pz2p⊥ observed observed C3CCC3 fsbfsb=0.5 σ 6.55 3.70 1.544 (diamond) 2s is promoted to 2p C fsb⨯fsb=0.5 calculated calculated (s⇒p) completely 3.71 1.541 fact2s,2p(C)=1.055 CC/ all bonding (0.125) σ fsbfsb=0.667 observed observed 2 C2CCC2 2s 2pz2p⊥ partial dual σ when 7.50 (graphite) 4.95 1.421 any not bonding(0.875) s⇏p calculated calculated 2s2pz2p⊥2p (all bonding) on both in-plane 1.421 sides 4.79 fact2s,2p(C)=1.055 when s⇒p on both sides. (0.875 0.875) bonding is reduced by coincident bonding to other Cs a D0 is the molecular bond dissociation energy. b Bond energies are from (or calculated from data) in W.M. Hayes, Editor in Chief, CRC Handbook of Chemistry and Physics, Section 5, CRC Press (2014). c Structural data and dipole moment () data are from W.M. Hayes, Editor in Chief, CRC Handbook of Chemistry and Physics, Section 9, CRC Press (2014). d fact is the orbital scale factor. Table IX. Components of the C2 Calculation at 1.243 Å (Energy in electron volts) 2 2 2 The configuration of C in C2 is 1s 2s 2pxy . Component Value Energy (eV) Increment 20 Sigma overlap of one pair of orbitals (overlap s_s) 0.7071 (There are two pairs of 2s bonding orbitals) Fraction_bonding s_s 0.4142 fraction_bonding s_s = (overlap s_s /(1+ overlap s_s) Simultaneous overlap of both pairs of orbitals (overlap s-s/s-s) 0.2071 fraction_bonding s_s fraction_bonding s_s /((1- fraction_bonding s_s fraction_bonding s_s) Total overlap of both pairs (overlap) 1.0000 overlap = 2 overlap s_s - 2 overlap s-s/s-s Fraction_bonding of both pairs 0.5000 fraction_bonding = (overlap/(1+ overlap) Kinetic energy of bond of one sigma orbital pair (2 KEbond s-s) 5.9265 KEbond s-s = Fraction_bonding ( KEsl+sr- KEsl - KEsr) Sigma bonding kinetic energy of both pairs 7.1539 7.1539 ( overlap/(1+overlap))/ (overlap s_s /(1+ overlap s_s)) 2 KEbond s-s Kinetic energy of 2 pi bonds 4 KEbond_ 4.2670 4.2670 Energy to make sigma pair 2s orthogonal to opposite 1s2 -6.4001 2 (KE - KE) where is 2s made orthogonal, is 2s Energy to make 1st pair orthogonal to opposite 1s2 when not bonding -3.2001 -3.2001 2 (KE - KE) (1- fraction_bonding) Additional Energy to make 2nd pair orthogonal to opposite 1s2 -1.8745 -1.8745 when bonding and 2nd pair not bonding 2 (KE - KE) fraction_bonding (1- fraction_bonding s_s) Nuclear-nuclear repulsion energy -416.9974 -416.9974 Nuclear to opposite two 2s sigma orbitals energy 236.0087 236.0087 When not bonding, (1-fraction_bonding), one 2s on each atom becomes an opposing orbital. Nuclear to opposite 1s orbitals energy 277.9988 277.9988 Nuclear to opposite opposing orbitals energy 50.4958 50.4958 Opposing orbitals occur only when not bonding, (1-fraction_bonding). The opposing orbital coefficients fpo and fso are chosen to make the orbital orthogonal to the opposite bonding 2s orbital. fsor overlap s_s_n - fpor overlap s_pz_n = 0.0 The overlap s_s_n is the overlap of the (not core orthogonalized) 2s atomic orbitals. Nuclear to opposite pxy orbitals energy 249.6900 249.6900 Electron-electron repulsion energy -392.2061 -392.2061 Energy to raise s to p -4.50 -4.50 2 (1- (overlap/(1+overlap))) fpo fpo 6.55+ 2 (1- (overlap/(1+overlap))) fpb fpb 7.90+ 4 (overlap/(1+overlap)) fpb fpb 7.90 21 Energy for C atom compression (scale factor 1.035) -0.33 -0.33 Total 6.51 Table X. Components of the CH4 Calculation at 1.106 Å (Energy in electron volts) 2 2 The configuration of C in H4C is 1s 2s2pz2pxy . CH4 has a tetrahedral structure. Component Value Overlap Energy change (eV) change Primary H – C Bond Sigma overlap of primary orbitals (overlap1st) 0.8814 0.8814 overlap 1st = 0.5 overlap s-s + 0.5 overlap s-p fsbr = fpbr = 0.5 Fraction_bonding (fraction_bonding1st) 0.4685 fraction_bonding1st = overlap1st /(1+ overlap1st) Kinetic energy of sigma orbital pair (2 KEbond) 6.5892 6.5892 KEbond = (1 /(1+ overlap1st)) (fsbr overlap s-s KEnet s-s+ fpbr overlap s-p KEnet s-pz ) Energy to make H 1s orthogonal to opposite C 1s2 -9.2300 (KE- KE) where is H 1s made orthogonal, is H 1s Energy to make H 1s orthogonal to opposite 1s2 when not bonding -4.6150 -4.6150 KEcore_ortho = (KE - KE) (1- fraction_bonding1st) Nuclear-nuclear repulsion energy -78.1189 -78.1189 C nuclear to H 1s energy 82.4697 82.4697 H nuclear to opposite two C 1s orbitals energy 26.0397 26.0397 H nuclear to opposite 2s orbital energy 12.6635 12.6635 H nuclear to opposite 2pz orbital energy 14.9262 14.9262 H nuclear to opposite two 2pxy orbital energy 22.8374 22.8374 H electron-C electron repulsions energy -76.8004 -76.8004 Energy to compress/polarize H -0.558 -0.558 Energy to raise C 2s to 2p -1.63 -1.63 Energy to compress C orbitals (orbital scale factor = 1.05) -0.08 -0.08 Total primary energy 3.723 Secondary H – H Bond 22 Secondary H 1s – H 1s overlap (secondary overlap s-s) 0.2605 Secondary H – H sigma overlap along primary axis (overlap 2nd) 0.1737 3 secondary atoms are at 1.8060 Å angle from bond axis is 35.265 degrees cos(35.265)=0.8165 overlap 2nd = 0.8165 0.8165 overlap s-s Secondary fraction_bonding (fraction_bonding 2nd) 0.1480 fraction_bonding 2nd = overlap 2nd /(1+overlap 2nd) First secondary fraction_bonding increment (fraction_bonding2nd_inc) 0.0393 fraction_bonding2nd_inc = 0.5 (1- fraction_bonding 1st) fraction_bonding2nd Secondary fraction_bonding is reduced by previous, coincident 1st order bonding. 0.5 because secondary spans 2 primary First secondary overlap increment 0.0409 0.0409 first secondary overlap = fraction_bonding2nd_inc/(1- fraction_bonding 2nd_inc) Second secondary fraction_bonding reduced 0.4798 by previous fraction_bonding of both primary and 1st secondary. fraction_bonding previous = (overlap 1st+ 0.0409)/(1+overlap 1st+ 0.0409) Second secondary overlap increment 0.0400 0.0400 second secondary overlap = fraction_bonding previous /(1- fraction_bonding previous) Third secondary overlap increment 0.0392 0.0392 Kinetic energy of first H – H bond (KEbond_2nd) 0.3112 KEbond_2nd= fraction_bonding 2nd ( KEsl+sr- KEsl - KEsr) First secondary kinetic energy increment (KEbond_2nd_inc) 0.1654 0.1654 KEbond_2nd_inc = (1- fraction_bonding 1st) KEbond_2nd Secondary KEbond_2nd is reduced by previous, coincident 1st order bonding. Second secondary kinetic energy increment (KEbond_2nd_inc) 0.1586 0.1586 Third secondary kinetic energy increment (KEbond_2nd_inc) 0.1524 0.1524 Single H nuclear-H nuclear repulsion energy -7.9730 Total nuclear-nuclear repulsion energy -11.9595 -11.9595 3 0.5 single nuclear-nuclear repulsion energy 0.5 because spans two primary bonds Total H nuclear to opposite H 1s (1.5 single nuclear to opposite) 24.8811 24.8811 Total H 1s to H 1s electron - electron repulsion (1.5 single ) -12.7765 -12.7765 Total secondary energy 0.6215 Totals 1.002 4.34 23 VIII. OTHER RESULTS The author has obtained results for many other molecules and lithium and beryllium metal. The additional results would require a discussion of sigma resonance, sigma/pi resonance, partial resonance, the impact of resonance on overlap, the calculation of fsb fsb for central atoms with bonds that are not identical (e.g. C in H3CCH3), additional discussion of secondary bonding, and other advanced topics which are beyond the scope of the present paper. The results presented herein are representative. IX. CONCLUSIONS The results presented herein support a model for the chemical bond which recognizes that the two electrons of the bonding pair are not completely identifiable, with respect to their source atom, when their orbitals overlap. The method for the calculation of the chemical bond, presented here, provides reasonably good results, appears to be general, and is not computationally intensive (A run of six bond lengths takes a few seconds on a desktop PC.). Particularly notable is the discovery that, except for the H2 case, the ratio overlap/ (1+ overlap) is 0.500 at the observed bond length. It is particularly hard to argue that this statistically sensible and computationally uncomplicated result is somehow fortuitous. It is also notable that the method described here is executed in terms understandable to the typical chemist (sigma bonds, pi bonds, hybrid orbitals, resonance, etc.). 1 C. A. Coulson, Trans. Faraday Soc., 33, 1479(1937). 2 For an overview of MO and other methods see Ira N. Levine, Quantum Chemistry 7th edition, Pearson (2014) or John P. Lowe, Kirk A. Peterson, Quantum Chemistry 3nd edition, Elsevier Academic Press (2006). 3 See, for example, Ira N. Levine, Quantum Chemistry 7th edition, pp 252, Pearson (2014). 4 C.A. Coulson, Valence 2nd edition, pp 91, Oxford University Press (1961). 5 W.E. Duncanson and C.A. Coulson, Proc.Roy.Soc. (Edinburgh), 62, 37 (1944). 6 See, for example, W.M. Hayes, Editor-In-Chief, CRC Handbook of Chemistry and Physics 95th Edition, Section 10 pp 10-197, CRC Press (2014). 7 See, for example, W.M. Hayes, Editor-In-Chief, CRC Handbook of Chemistry and Physics 95th Edition, Section 10 pp 10-147 CRC Press (2014). 24